EL0011

euclidean_gcd_execution_logarithmic_exists

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

For every pair of naturals construct all witnesses at once: canonical bit length, an actual terminal-gcd Euclidean beta execution, and the exact logarithmic step bound 2*BitLen(b)+1.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Exact expanded first-order arithmetic statement

forall a b. exists l g k. ((((((b) = 0 /\ (l) = 1) \/ exists ff_exponent_bl_elb_root_length ff_lower_bl_elb_root_length ff_upper_bl_elb_root_length. (((l) = S ff_exponent_bl_elb_root_length) /\ ((exists ff_positive_bl_elb_root_length. ff_positive_bl_elb_root_length + 1 = (b)) /\ ((exists pa_b_bl_elb_root_length_lower pa_c_bl_elb_root_length_lower. ((forall pa_i_bl_elb_root_length_lower_repeat. (exists pa_lt_bl_elb_root_length_lower_repeat_bound. pa_lt_bl_elb_root_length_lower_repeat_bound + S pa_i_bl_elb_root_length_lower_repeat = ff_exponent_bl_elb_root_length) -> (((exists pa_h_bl_elb_root_length_lower_repeat_decoded. pa_h_bl_elb_root_length_lower_repeat_decoded + S (2) = S ((S (pa_i_bl_elb_root_length_lower_repeat)) * pa_c_bl_elb_root_length_lower)) /\ exists pa_q_bl_elb_root_length_lower_repeat_decoded. pa_b_bl_elb_root_length_lower = pa_q_bl_elb_root_length_lower_repeat_decoded * S ((S (pa_i_bl_elb_root_length_lower_repeat)) * pa_c_bl_elb_root_length_lower) + (2)))) /\ (exists pa_u_bl_elb_root_length_lower_product pa_v_bl_elb_root_length_lower_product. ((((exists pa_h_bl_elb_root_length_lower_product_start. pa_h_bl_elb_root_length_lower_product_start + S (1) = S ((S (0)) * pa_v_bl_elb_root_length_lower_product)) /\ exists pa_q_bl_elb_root_length_lower_product_start. pa_u_bl_elb_root_length_lower_product = pa_q_bl_elb_root_length_lower_product_start * S ((S (0)) * pa_v_bl_elb_root_length_lower_product) + (1))) /\ ((((exists pa_h_bl_elb_root_length_lower_product_terminal. pa_h_bl_elb_root_length_lower_product_terminal + S (ff_lower_bl_elb_root_length) = S ((S (ff_exponent_bl_elb_root_length)) * pa_v_bl_elb_root_length_lower_product)) /\ exists pa_q_bl_elb_root_length_lower_product_terminal. pa_u_bl_elb_root_length_lower_product = pa_q_bl_elb_root_length_lower_product_terminal * S ((S (ff_exponent_bl_elb_root_length)) * pa_v_bl_elb_root_length_lower_product) + (ff_lower_bl_elb_root_length))) /\ forall pa_i_bl_elb_root_length_lower_product. (exists pa_lt_bl_elb_root_length_lower_product_bound. pa_lt_bl_elb_root_length_lower_product_bound + S pa_i_bl_elb_root_length_lower_product = ff_exponent_bl_elb_root_length) -> exists pa_p_bl_elb_root_length_lower_product pa_r_bl_elb_root_length_lower_product pa_s_bl_elb_root_length_lower_product. ((((exists pa_h_bl_elb_root_length_lower_product_factor. pa_h_bl_elb_root_length_lower_product_factor + S (pa_p_bl_elb_root_length_lower_product) = S ((S (pa_i_bl_elb_root_length_lower_product)) * pa_c_bl_elb_root_length_lower)) /\ exists pa_q_bl_elb_root_length_lower_product_factor. pa_b_bl_elb_root_length_lower = pa_q_bl_elb_root_length_lower_product_factor * S ((S (pa_i_bl_elb_root_length_lower_product)) * pa_c_bl_elb_root_length_lower) + (pa_p_bl_elb_root_length_lower_product))) /\ ((((exists pa_h_bl_elb_root_length_lower_product_partial. pa_h_bl_elb_root_length_lower_product_partial + S (pa_r_bl_elb_root_length_lower_product) = S ((S (pa_i_bl_elb_root_length_lower_product)) * pa_v_bl_elb_root_length_lower_product)) /\ exists pa_q_bl_elb_root_length_lower_product_partial. pa_u_bl_elb_root_length_lower_product = pa_q_bl_elb_root_length_lower_product_partial * S ((S (pa_i_bl_elb_root_length_lower_product)) * pa_v_bl_elb_root_length_lower_product) + (pa_r_bl_elb_root_length_lower_product))) /\ ((((exists pa_h_bl_elb_root_length_lower_product_successor. pa_h_bl_elb_root_length_lower_product_successor + S (pa_s_bl_elb_root_length_lower_product) = S ((S (S pa_i_bl_elb_root_length_lower_product)) * pa_v_bl_elb_root_length_lower_product)) /\ exists pa_q_bl_elb_root_length_lower_product_successor. pa_u_bl_elb_root_length_lower_product = pa_q_bl_elb_root_length_lower_product_successor * S ((S (S pa_i_bl_elb_root_length_lower_product)) * pa_v_bl_elb_root_length_lower_product) + (pa_s_bl_elb_root_length_lower_product))) /\ pa_s_bl_elb_root_length_lower_product = pa_r_bl_elb_root_length_lower_product * pa_p_bl_elb_root_length_lower_product)))))))) /\ ((exists pa_b_bl_elb_root_length_upper pa_c_bl_elb_root_length_upper. ((forall pa_i_bl_elb_root_length_upper_repeat. (exists pa_lt_bl_elb_root_length_upper_repeat_bound. pa_lt_bl_elb_root_length_upper_repeat_bound + S pa_i_bl_elb_root_length_upper_repeat = l) -> (((exists pa_h_bl_elb_root_length_upper_repeat_decoded. pa_h_bl_elb_root_length_upper_repeat_decoded + S (2) = S ((S (pa_i_bl_elb_root_length_upper_repeat)) * pa_c_bl_elb_root_length_upper)) /\ exists pa_q_bl_elb_root_length_upper_repeat_decoded. pa_b_bl_elb_root_length_upper = pa_q_bl_elb_root_length_upper_repeat_decoded * S ((S (pa_i_bl_elb_root_length_upper_repeat)) * pa_c_bl_elb_root_length_upper) + (2)))) /\ (exists pa_u_bl_elb_root_length_upper_product pa_v_bl_elb_root_length_upper_product. ((((exists pa_h_bl_elb_root_length_upper_product_start. pa_h_bl_elb_root_length_upper_product_start + S (1) = S ((S (0)) * pa_v_bl_elb_root_length_upper_product)) /\ exists pa_q_bl_elb_root_length_upper_product_start. pa_u_bl_elb_root_length_upper_product = pa_q_bl_elb_root_length_upper_product_start * S ((S (0)) * pa_v_bl_elb_root_length_upper_product) + (1))) /\ ((((exists pa_h_bl_elb_root_length_upper_product_terminal. pa_h_bl_elb_root_length_upper_product_terminal + S (ff_upper_bl_elb_root_length) = S ((S (l)) * pa_v_bl_elb_root_length_upper_product)) /\ exists pa_q_bl_elb_root_length_upper_product_terminal. pa_u_bl_elb_root_length_upper_product = pa_q_bl_elb_root_length_upper_product_terminal * S ((S (l)) * pa_v_bl_elb_root_length_upper_product) + (ff_upper_bl_elb_root_length))) /\ forall pa_i_bl_elb_root_length_upper_product. (exists pa_lt_bl_elb_root_length_upper_product_bound. pa_lt_bl_elb_root_length_upper_product_bound + S pa_i_bl_elb_root_length_upper_product = l) -> exists pa_p_bl_elb_root_length_upper_product pa_r_bl_elb_root_length_upper_product pa_s_bl_elb_root_length_upper_product. ((((exists pa_h_bl_elb_root_length_upper_product_factor. pa_h_bl_elb_root_length_upper_product_factor + S (pa_p_bl_elb_root_length_upper_product) = S ((S (pa_i_bl_elb_root_length_upper_product)) * pa_c_bl_elb_root_length_upper)) /\ exists pa_q_bl_elb_root_length_upper_product_factor. pa_b_bl_elb_root_length_upper = pa_q_bl_elb_root_length_upper_product_factor * S ((S (pa_i_bl_elb_root_length_upper_product)) * pa_c_bl_elb_root_length_upper) + (pa_p_bl_elb_root_length_upper_product))) /\ ((((exists pa_h_bl_elb_root_length_upper_product_partial. pa_h_bl_elb_root_length_upper_product_partial + S (pa_r_bl_elb_root_length_upper_product) = S ((S (pa_i_bl_elb_root_length_upper_product)) * pa_v_bl_elb_root_length_upper_product)) /\ exists pa_q_bl_elb_root_length_upper_product_partial. pa_u_bl_elb_root_length_upper_product = pa_q_bl_elb_root_length_upper_product_partial * S ((S (pa_i_bl_elb_root_length_upper_product)) * pa_v_bl_elb_root_length_upper_product) + (pa_r_bl_elb_root_length_upper_product))) /\ ((((exists pa_h_bl_elb_root_length_upper_product_successor. pa_h_bl_elb_root_length_upper_product_successor + S (pa_s_bl_elb_root_length_upper_product) = S ((S (S pa_i_bl_elb_root_length_upper_product)) * pa_v_bl_elb_root_length_upper_product)) /\ exists pa_q_bl_elb_root_length_upper_product_successor. pa_u_bl_elb_root_length_upper_product = pa_q_bl_elb_root_length_upper_product_successor * S ((S (S pa_i_bl_elb_root_length_upper_product)) * pa_v_bl_elb_root_length_upper_product) + (pa_s_bl_elb_root_length_upper_product))) /\ pa_s_bl_elb_root_length_upper_product = pa_r_bl_elb_root_length_upper_product * pa_p_bl_elb_root_length_upper_product)))))))) /\ ((exists ff_lower_gap_bl_elb_root_length. ff_lower_gap_bl_elb_root_length + (ff_lower_bl_elb_root_length) = (b)) /\ (exists ff_upper_gap_bl_elb_root_length. ff_upper_gap_bl_elb_root_length + S (b) = (ff_upper_bl_elb_root_length))))))))) /\ ((exists egt_list_elb_root_execution egt_history_elb_root_execution egt_scale_elb_root_execution. ((exists cf_gcd_egt_elb_root_execution_trace. ((((exists ff_h_cf_egt_elb_root_execution_trace_initial_state. ff_h_cf_egt_elb_root_execution_trace_initial_state + S (((cf_gcd_egt_elb_root_execution_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_egt_elb_root_execution_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) + ((((0) + (0)) * S ((0) + (0)) + ((0) + (0))) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0))))) = S ((S (0)) * egt_scale_elb_root_execution)) /\ exists ff_q_cf_egt_elb_root_execution_trace_initial_state. egt_history_elb_root_execution = ff_q_cf_egt_elb_root_execution_trace_initial_state * S ((S (0)) * egt_scale_elb_root_execution) + (((cf_gcd_egt_elb_root_execution_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_egt_elb_root_execution_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) + ((((0) + (0)) * S ((0) + (0)) + ((0) + (0))) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0))))))) /\ ((((exists ff_h_cf_egt_elb_root_execution_trace_terminal_state. ff_h_cf_egt_elb_root_execution_trace_terminal_state + S (((a) + (((b) + (egt_list_elb_root_execution)) * S ((b) + (egt_list_elb_root_execution)) + ((egt_list_elb_root_execution) + (egt_list_elb_root_execution)))) * S ((a) + (((b) + (egt_list_elb_root_execution)) * S ((b) + (egt_list_elb_root_execution)) + ((egt_list_elb_root_execution) + (egt_list_elb_root_execution)))) + ((((b) + (egt_list_elb_root_execution)) * S ((b) + (egt_list_elb_root_execution)) + ((egt_list_elb_root_execution) + (egt_list_elb_root_execution))) + (((b) + (egt_list_elb_root_execution)) * S ((b) + (egt_list_elb_root_execution)) + ((egt_list_elb_root_execution) + (egt_list_elb_root_execution))))) = S ((S (k)) * egt_scale_elb_root_execution)) /\ exists ff_q_cf_egt_elb_root_execution_trace_terminal_state. egt_history_elb_root_execution = ff_q_cf_egt_elb_root_execution_trace_terminal_state * S ((S (k)) * egt_scale_elb_root_execution) + (((a) + (((b) + (egt_list_elb_root_execution)) * S ((b) + (egt_list_elb_root_execution)) + ((egt_list_elb_root_execution) + (egt_list_elb_root_execution)))) * S ((a) + (((b) + (egt_list_elb_root_execution)) * S ((b) + (egt_list_elb_root_execution)) + ((egt_list_elb_root_execution) + (egt_list_elb_root_execution)))) + ((((b) + (egt_list_elb_root_execution)) * S ((b) + (egt_list_elb_root_execution)) + ((egt_list_elb_root_execution) + (egt_list_elb_root_execution))) + (((b) + (egt_list_elb_root_execution)) * S ((b) + (egt_list_elb_root_execution)) + ((egt_list_elb_root_execution) + (egt_list_elb_root_execution))))))) /\ forall cf_index_egt_elb_root_execution_trace. (exists ff_lt_cf_egt_elb_root_execution_trace_index. ff_lt_cf_egt_elb_root_execution_trace_index + S cf_index_egt_elb_root_execution_trace = k) -> exists cf_old_a_egt_elb_root_execution_trace cf_old_b_egt_elb_root_execution_trace cf_tail_egt_elb_root_execution_trace cf_new_a_egt_elb_root_execution_trace cf_new_b_egt_elb_root_execution_trace cf_head_egt_elb_root_execution_trace cf_quotient_egt_elb_root_execution_trace. ((((exists ff_h_cf_egt_elb_root_execution_trace_previous_state. ff_h_cf_egt_elb_root_execution_trace_previous_state + S (((cf_old_a_egt_elb_root_execution_trace) + (((cf_old_b_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)) * S ((cf_old_b_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)) + ((cf_tail_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)))) * S ((cf_old_a_egt_elb_root_execution_trace) + (((cf_old_b_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)) * S ((cf_old_b_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)) + ((cf_tail_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)))) + ((((cf_old_b_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)) * S ((cf_old_b_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)) + ((cf_tail_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace))) + (((cf_old_b_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)) * S ((cf_old_b_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)) + ((cf_tail_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace))))) = S ((S (cf_index_egt_elb_root_execution_trace)) * egt_scale_elb_root_execution)) /\ exists ff_q_cf_egt_elb_root_execution_trace_previous_state. egt_history_elb_root_execution = ff_q_cf_egt_elb_root_execution_trace_previous_state * S ((S (cf_index_egt_elb_root_execution_trace)) * egt_scale_elb_root_execution) + (((cf_old_a_egt_elb_root_execution_trace) + (((cf_old_b_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)) * S ((cf_old_b_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)) + ((cf_tail_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)))) * S ((cf_old_a_egt_elb_root_execution_trace) + (((cf_old_b_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)) * S ((cf_old_b_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)) + ((cf_tail_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)))) + ((((cf_old_b_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)) * S ((cf_old_b_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)) + ((cf_tail_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace))) + (((cf_old_b_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)) * S ((cf_old_b_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace)) + ((cf_tail_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace))))))) /\ ((((exists ff_h_cf_egt_elb_root_execution_trace_following_state. ff_h_cf_egt_elb_root_execution_trace_following_state + S (((cf_new_a_egt_elb_root_execution_trace) + (((cf_new_b_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)) * S ((cf_new_b_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)) + ((cf_head_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)))) * S ((cf_new_a_egt_elb_root_execution_trace) + (((cf_new_b_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)) * S ((cf_new_b_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)) + ((cf_head_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)))) + ((((cf_new_b_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)) * S ((cf_new_b_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)) + ((cf_head_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace))) + (((cf_new_b_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)) * S ((cf_new_b_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)) + ((cf_head_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace))))) = S ((S (S cf_index_egt_elb_root_execution_trace)) * egt_scale_elb_root_execution)) /\ exists ff_q_cf_egt_elb_root_execution_trace_following_state. egt_history_elb_root_execution = ff_q_cf_egt_elb_root_execution_trace_following_state * S ((S (S cf_index_egt_elb_root_execution_trace)) * egt_scale_elb_root_execution) + (((cf_new_a_egt_elb_root_execution_trace) + (((cf_new_b_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)) * S ((cf_new_b_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)) + ((cf_head_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)))) * S ((cf_new_a_egt_elb_root_execution_trace) + (((cf_new_b_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)) * S ((cf_new_b_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)) + ((cf_head_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)))) + ((((cf_new_b_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)) * S ((cf_new_b_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)) + ((cf_head_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace))) + (((cf_new_b_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)) * S ((cf_new_b_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace)) + ((cf_head_egt_elb_root_execution_trace) + (cf_head_egt_elb_root_execution_trace))))))) /\ (cf_new_b_egt_elb_root_execution_trace = cf_old_a_egt_elb_root_execution_trace /\ (cf_new_a_egt_elb_root_execution_trace = cf_new_b_egt_elb_root_execution_trace * cf_quotient_egt_elb_root_execution_trace + cf_old_b_egt_elb_root_execution_trace /\ ((exists ff_lt_cf_egt_elb_root_execution_trace_remainder. ff_lt_cf_egt_elb_root_execution_trace_remainder + S cf_old_b_egt_elb_root_execution_trace = cf_new_b_egt_elb_root_execution_trace) /\ (cf_head_egt_elb_root_execution_trace = S ((cf_quotient_egt_elb_root_execution_trace + cf_tail_egt_elb_root_execution_trace) * S (cf_quotient_egt_elb_root_execution_trace + cf_tail_egt_elb_root_execution_trace) + (cf_tail_egt_elb_root_execution_trace + cf_tail_egt_elb_root_execution_trace))))))))))) /\ ((((exists ff_h_cf_egt_elb_root_execution_initial_state. ff_h_cf_egt_elb_root_execution_initial_state + S (((g) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((g) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) + ((((0) + (0)) * S ((0) + (0)) + ((0) + (0))) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0))))) = S ((S (0)) * egt_scale_elb_root_execution)) /\ exists ff_q_cf_egt_elb_root_execution_initial_state. egt_history_elb_root_execution = ff_q_cf_egt_elb_root_execution_initial_state * S ((S (0)) * egt_scale_elb_root_execution) + (((g) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((g) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) + ((((0) + (0)) * S ((0) + (0)) + ((0) + (0))) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0))))))) /\ ((((exists ec_gcd_left_egt_elb_root_execution_result. a = g * ec_gcd_left_egt_elb_root_execution_result) /\ (exists ec_gcd_right_egt_elb_root_execution_result. b = g * ec_gcd_right_egt_elb_root_execution_result)) /\ forall ec_gcd_common_egt_elb_root_execution_result. (exists ec_gcd_common_left_egt_elb_root_execution_result. a = ec_gcd_common_egt_elb_root_execution_result * ec_gcd_common_left_egt_elb_root_execution_result) -> (exists ec_gcd_common_right_egt_elb_root_execution_result. b = ec_gcd_common_egt_elb_root_execution_result * ec_gcd_common_right_egt_elb_root_execution_result) -> exists ec_gcd_greatest_egt_elb_root_execution_result. g = ec_gcd_common_egt_elb_root_execution_result * ec_gcd_greatest_egt_elb_root_execution_result))))) /\ exists elb_bound_gap_root. elb_bound_gap_root + k = 2 * l + 1)))

Constructive proof overview

Generated structural guide

For every pair of naturals construct all witnesses at once: canonical bit length, an actual terminal-gcd Euclidean beta execution, and the exact logarithmic step bound 2*BitLen(b)+1.

The unchanged tactic script uses 2 declared prerequisites and contains 21 exact native proof lines.

Alpha v34 checked-use · first admitted v23 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

binary_length_exists Alpha theorem; checked-use authorized EL0010 euclidean_gcd_execution_logarithmic_bound

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

21 script commands · 11 reading checkpoints · 1 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Named ingredients (1)
euclidean gcd execution logarithmic bound

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–2

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro a
  2. L2
    intro b
02Use earlier factsL3–3

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L3
    specialize binary_length_exists b
03Separate the logical casesL4–4

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L4
    cases binary_length_exists
04Use earlier factsL5–7

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L5
    specialize euclidean_gcd_execution_logarithmic_bound a
  2. L6
    specialize euclidean_gcd_execution_logarithmic_bound b
  3. L7
    specialize euclidean_gcd_execution_logarithmic_bound x
05Establish hexecutionL8–10

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply euclidean gcd execution logarithmic bound.

  1. L8
    have hexecution : ∃ g. ∃ k. EuclideanAnchoredExecution(a,b,g,k) ∧ Le(k,2 · x + 1)Definitions: EuclideanAnchoredExecutionLe
  2. L9
    apply euclidean_gcd_execution_logarithmic_bound
  3. L10
    exact binary_length_exists_witness
06Separate the logical casesL11–13

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L11
    cases hexecution
  2. L12
    cases hexecution_witness
  3. L13
    cases hexecution_witness_witness
07Construct an explicit witnessL14–16

Supply the displayed value, then prove that it has the required property.

  1. L14
    exists x
  2. L15
    exists x1
  3. L16
    exists x2
08Separate the logical casesL17–17

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L17
    split
09Use earlier factsL18–18

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L18
    exact binary_length_exists_witness
10Separate the logical casesL19–19

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L19
    split
11Use earlier factsL20–21

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L20
    exact hexecution_witness_witness_left
  2. L21
    exact hexecution_witness_witness_right

Library-wide reading audit

Original exact command ledger · 21 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003specialize binary_length_exists b
  4. 0004cases binary_length_exists
  5. 0005specialize euclidean_gcd_execution_logarithmic_bound a
  6. 0006specialize euclidean_gcd_execution_logarithmic_bound b
  7. 0007specialize euclidean_gcd_execution_logarithmic_bound x
  8. 0008have hexecution : exists g k. ((exists egt_list_elb_total egt_history_elb_total egt_scale_elb_total. ((exists cf_gcd_egt_elb_total_trace. ((((exists ff_h_cf_egt_elb_total_trace_initial_state. ff_h_cf_egt_elb_total_trace_initial_state + S (((cf_gcd_egt_elb_total_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_egt_elb_total_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) + ((((0) + (0)) * S ((0) + (0)) + ((0) + (0))) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0))))) = S ((S (0)) * egt_scale_elb_total)) /\ exists ff_q_cf_egt_elb_total_trace_initial_state. egt_history_elb_total = ff_q_cf_egt_elb_total_trace_initial_state * S ((S (0)) * egt_scale_elb_total) + (((cf_gcd_egt_elb_total_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_egt_elb_total_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) + ((((0) + (0)) * S ((0) + (0)) + ((0) + (0))) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0))))))) /\ ((((exists ff_h_cf_egt_elb_total_trace_terminal_state. ff_h_cf_egt_elb_total_trace_terminal_state + S (((a) + (((b) + (egt_list_elb_total)) * S ((b) + (egt_list_elb_total)) + ((egt_list_elb_total) + (egt_list_elb_total)))) * S ((a) + (((b) + (egt_list_elb_total)) * S ((b) + (egt_list_elb_total)) + ((egt_list_elb_total) + (egt_list_elb_total)))) + ((((b) + (egt_list_elb_total)) * S ((b) + (egt_list_elb_total)) + ((egt_list_elb_total) + (egt_list_elb_total))) + (((b) + (egt_list_elb_total)) * S ((b) + (egt_list_elb_total)) + ((egt_list_elb_total) + (egt_list_elb_total))))) = S ((S (k)) * egt_scale_elb_total)) /\ exists ff_q_cf_egt_elb_total_trace_terminal_state. egt_history_elb_total = ff_q_cf_egt_elb_total_trace_terminal_state * S ((S (k)) * egt_scale_elb_total) + (((a) + (((b) + (egt_list_elb_total)) * S ((b) + (egt_list_elb_total)) + ((egt_list_elb_total) + (egt_list_elb_total)))) * S ((a) + (((b) + (egt_list_elb_total)) * S ((b) + (egt_list_elb_total)) + ((egt_list_elb_total) + (egt_list_elb_total)))) + ((((b) + (egt_list_elb_total)) * S ((b) + (egt_list_elb_total)) + ((egt_list_elb_total) + (egt_list_elb_total))) + (((b) + (egt_list_elb_total)) * S ((b) + (egt_list_elb_total)) + ((egt_list_elb_total) + (egt_list_elb_total))))))) /\ forall cf_index_egt_elb_total_trace. (exists ff_lt_cf_egt_elb_total_trace_index. ff_lt_cf_egt_elb_total_trace_index + S cf_index_egt_elb_total_trace = k) -> exists cf_old_a_egt_elb_total_trace cf_old_b_egt_elb_total_trace cf_tail_egt_elb_total_trace cf_new_a_egt_elb_total_trace cf_new_b_egt_elb_total_trace cf_head_egt_elb_total_trace cf_quotient_egt_elb_total_trace. ((((exists ff_h_cf_egt_elb_total_trace_previous_state. ff_h_cf_egt_elb_total_trace_previous_state + S (((cf_old_a_egt_elb_total_trace) + (((cf_old_b_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)) * S ((cf_old_b_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)) + ((cf_tail_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)))) * S ((cf_old_a_egt_elb_total_trace) + (((cf_old_b_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)) * S ((cf_old_b_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)) + ((cf_tail_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)))) + ((((cf_old_b_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)) * S ((cf_old_b_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)) + ((cf_tail_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace))) + (((cf_old_b_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)) * S ((cf_old_b_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)) + ((cf_tail_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace))))) = S ((S (cf_index_egt_elb_total_trace)) * egt_scale_elb_total)) /\ exists ff_q_cf_egt_elb_total_trace_previous_state. egt_history_elb_total = ff_q_cf_egt_elb_total_trace_previous_state * S ((S (cf_index_egt_elb_total_trace)) * egt_scale_elb_total) + (((cf_old_a_egt_elb_total_trace) + (((cf_old_b_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)) * S ((cf_old_b_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)) + ((cf_tail_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)))) * S ((cf_old_a_egt_elb_total_trace) + (((cf_old_b_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)) * S ((cf_old_b_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)) + ((cf_tail_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)))) + ((((cf_old_b_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)) * S ((cf_old_b_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)) + ((cf_tail_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace))) + (((cf_old_b_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)) * S ((cf_old_b_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace)) + ((cf_tail_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace))))))) /\ ((((exists ff_h_cf_egt_elb_total_trace_following_state. ff_h_cf_egt_elb_total_trace_following_state + S (((cf_new_a_egt_elb_total_trace) + (((cf_new_b_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)) * S ((cf_new_b_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)) + ((cf_head_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)))) * S ((cf_new_a_egt_elb_total_trace) + (((cf_new_b_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)) * S ((cf_new_b_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)) + ((cf_head_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)))) + ((((cf_new_b_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)) * S ((cf_new_b_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)) + ((cf_head_egt_elb_total_trace) + (cf_head_egt_elb_total_trace))) + (((cf_new_b_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)) * S ((cf_new_b_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)) + ((cf_head_egt_elb_total_trace) + (cf_head_egt_elb_total_trace))))) = S ((S (S cf_index_egt_elb_total_trace)) * egt_scale_elb_total)) /\ exists ff_q_cf_egt_elb_total_trace_following_state. egt_history_elb_total = ff_q_cf_egt_elb_total_trace_following_state * S ((S (S cf_index_egt_elb_total_trace)) * egt_scale_elb_total) + (((cf_new_a_egt_elb_total_trace) + (((cf_new_b_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)) * S ((cf_new_b_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)) + ((cf_head_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)))) * S ((cf_new_a_egt_elb_total_trace) + (((cf_new_b_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)) * S ((cf_new_b_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)) + ((cf_head_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)))) + ((((cf_new_b_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)) * S ((cf_new_b_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)) + ((cf_head_egt_elb_total_trace) + (cf_head_egt_elb_total_trace))) + (((cf_new_b_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)) * S ((cf_new_b_egt_elb_total_trace) + (cf_head_egt_elb_total_trace)) + ((cf_head_egt_elb_total_trace) + (cf_head_egt_elb_total_trace))))))) /\ (cf_new_b_egt_elb_total_trace = cf_old_a_egt_elb_total_trace /\ (cf_new_a_egt_elb_total_trace = cf_new_b_egt_elb_total_trace * cf_quotient_egt_elb_total_trace + cf_old_b_egt_elb_total_trace /\ ((exists ff_lt_cf_egt_elb_total_trace_remainder. ff_lt_cf_egt_elb_total_trace_remainder + S cf_old_b_egt_elb_total_trace = cf_new_b_egt_elb_total_trace) /\ (cf_head_egt_elb_total_trace = S ((cf_quotient_egt_elb_total_trace + cf_tail_egt_elb_total_trace) * S (cf_quotient_egt_elb_total_trace + cf_tail_egt_elb_total_trace) + (cf_tail_egt_elb_total_trace + cf_tail_egt_elb_total_trace))))))))))) /\ ((((exists ff_h_cf_egt_elb_total_initial_state. ff_h_cf_egt_elb_total_initial_state + S (((g) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((g) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) + ((((0) + (0)) * S ((0) + (0)) + ((0) + (0))) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0))))) = S ((S (0)) * egt_scale_elb_total)) /\ exists ff_q_cf_egt_elb_total_initial_state. egt_history_elb_total = ff_q_cf_egt_elb_total_initial_state * S ((S (0)) * egt_scale_elb_total) + (((g) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((g) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) + ((((0) + (0)) * S ((0) + (0)) + ((0) + (0))) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0))))))) /\ ((((exists ec_gcd_left_egt_elb_total_result. a = g * ec_gcd_left_egt_elb_total_result) /\ (exists ec_gcd_right_egt_elb_total_result. b = g * ec_gcd_right_egt_elb_total_result)) /\ forall ec_gcd_common_egt_elb_total_result. (exists ec_gcd_common_left_egt_elb_total_result. a = ec_gcd_common_egt_elb_total_result * ec_gcd_common_left_egt_elb_total_result) -> (exists ec_gcd_common_right_egt_elb_total_result. b = ec_gcd_common_egt_elb_total_result * ec_gcd_common_right_egt_elb_total_result) -> exists ec_gcd_greatest_egt_elb_total_result. g = ec_gcd_common_egt_elb_total_result * ec_gcd_greatest_egt_elb_total_result))))) /\ exists gap. gap + k = 2 * x + 1)
  9. 0009apply euclidean_gcd_execution_logarithmic_bound
  10. 0010exact binary_length_exists_witness
  11. 0011cases hexecution
  12. 0012cases hexecution_witness
  13. 0013cases hexecution_witness_witness
  14. 0014exists x
  15. 0015exists x1
  16. 0016exists x2
  17. 0017split
  18. 0018exact binary_length_exists_witness
  19. 0019split
  20. 0020exact hexecution_witness_witness_left
  21. 0021exact hexecution_witness_witness_right

Separate complete second-wave branches: Full T13 proof · Alpha v27.