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.
Hygienic expanded first-order definition
exists ec_list_advanced ec_history_advanced ec_scale_advanced. ((exists cf_gcd_ec_advanced_trace. ((((exists ff_h_cf_ec_advanced_trace_initial_state. ff_h_cf_ec_advanced_trace_initial_state + S (((cf_gcd_ec_advanced_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_ec_advanced_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)) * ec_scale_advanced)) /\ exists ff_q_cf_ec_advanced_trace_initial_state. ec_history_advanced = ff_q_cf_ec_advanced_trace_initial_state * S ((S (0)) * ec_scale_advanced) + (((cf_gcd_ec_advanced_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_ec_advanced_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_ec_advanced_trace_terminal_state. ff_h_cf_ec_advanced_trace_terminal_state + S (((a) + (((b) + (ec_list_advanced)) * S ((b) + (ec_list_advanced)) + ((ec_list_advanced) + (ec_list_advanced)))) * S ((a) + (((b) + (ec_list_advanced)) * S ((b) + (ec_list_advanced)) + ((ec_list_advanced) + (ec_list_advanced)))) + ((((b) + (ec_list_advanced)) * S ((b) + (ec_list_advanced)) + ((ec_list_advanced) + (ec_list_advanced))) + (((b) + (ec_list_advanced)) * S ((b) + (ec_list_advanced)) + ((ec_list_advanced) + (ec_list_advanced))))) = S ((S (k)) * ec_scale_advanced)) /\ exists ff_q_cf_ec_advanced_trace_terminal_state. ec_history_advanced = ff_q_cf_ec_advanced_trace_terminal_state * S ((S (k)) * ec_scale_advanced) + (((a) + (((b) + (ec_list_advanced)) * S ((b) + (ec_list_advanced)) + ((ec_list_advanced) + (ec_list_advanced)))) * S ((a) + (((b) + (ec_list_advanced)) * S ((b) + (ec_list_advanced)) + ((ec_list_advanced) + (ec_list_advanced)))) + ((((b) + (ec_list_advanced)) * S ((b) + (ec_list_advanced)) + ((ec_list_advanced) + (ec_list_advanced))) + (((b) + (ec_list_advanced)) * S ((b) + (ec_list_advanced)) + ((ec_list_advanced) + (ec_list_advanced))))))) /\ forall cf_index_ec_advanced_trace. (exists ff_lt_cf_ec_advanced_trace_index. ff_lt_cf_ec_advanced_trace_index + S cf_index_ec_advanced_trace = k) -> exists cf_old_a_ec_advanced_trace cf_old_b_ec_advanced_trace cf_tail_ec_advanced_trace cf_new_a_ec_advanced_trace cf_new_b_ec_advanced_trace cf_head_ec_advanced_trace cf_quotient_ec_advanced_trace. ((((exists ff_h_cf_ec_advanced_trace_previous_state. ff_h_cf_ec_advanced_trace_previous_state + S (((cf_old_a_ec_advanced_trace) + (((cf_old_b_ec_advanced_trace) + (cf_tail_ec_advanced_trace)) * S ((cf_old_b_ec_advanced_trace) + (cf_tail_ec_advanced_trace)) + ((cf_tail_ec_advanced_trace) + (cf_tail_ec_advanced_trace)))) * S ((cf_old_a_ec_advanced_trace) + (((cf_old_b_ec_advanced_trace) + (cf_tail_ec_advanced_trace)) * S ((cf_old_b_ec_advanced_trace) + (cf_tail_ec_advanced_trace)) + ((cf_tail_ec_advanced_trace) + (cf_tail_ec_advanced_trace)))) + ((((cf_old_b_ec_advanced_trace) + (cf_tail_ec_advanced_trace)) * S ((cf_old_b_ec_advanced_trace) + (cf_tail_ec_advanced_trace)) + ((cf_tail_ec_advanced_trace) + (cf_tail_ec_advanced_trace))) + (((cf_old_b_ec_advanced_trace) + (cf_tail_ec_advanced_trace)) * S ((cf_old_b_ec_advanced_trace) + (cf_tail_ec_advanced_trace)) + ((cf_tail_ec_advanced_trace) + (cf_tail_ec_advanced_trace))))) = S ((S (cf_index_ec_advanced_trace)) * ec_scale_advanced)) /\ exists ff_q_cf_ec_advanced_trace_previous_state. ec_history_advanced = ff_q_cf_ec_advanced_trace_previous_state * S ((S (cf_index_ec_advanced_trace)) * ec_scale_advanced) + (((cf_old_a_ec_advanced_trace) + (((cf_old_b_ec_advanced_trace) + (cf_tail_ec_advanced_trace)) * S ((cf_old_b_ec_advanced_trace) + (cf_tail_ec_advanced_trace)) + ((cf_tail_ec_advanced_trace) + (cf_tail_ec_advanced_trace)))) * S ((cf_old_a_ec_advanced_trace) + (((cf_old_b_ec_advanced_trace) + (cf_tail_ec_advanced_trace)) * S ((cf_old_b_ec_advanced_trace) + (cf_tail_ec_advanced_trace)) + ((cf_tail_ec_advanced_trace) + (cf_tail_ec_advanced_trace)))) + ((((cf_old_b_ec_advanced_trace) + (cf_tail_ec_advanced_trace)) * S ((cf_old_b_ec_advanced_trace) + (cf_tail_ec_advanced_trace)) + ((cf_tail_ec_advanced_trace) + (cf_tail_ec_advanced_trace))) + (((cf_old_b_ec_advanced_trace) + (cf_tail_ec_advanced_trace)) * S ((cf_old_b_ec_advanced_trace) + (cf_tail_ec_advanced_trace)) + ((cf_tail_ec_advanced_trace) + (cf_tail_ec_advanced_trace))))))) /\ ((((exists ff_h_cf_ec_advanced_trace_following_state. ff_h_cf_ec_advanced_trace_following_state + S (((cf_new_a_ec_advanced_trace) + (((cf_new_b_ec_advanced_trace) + (cf_head_ec_advanced_trace)) * S ((cf_new_b_ec_advanced_trace) + (cf_head_ec_advanced_trace)) + ((cf_head_ec_advanced_trace) + (cf_head_ec_advanced_trace)))) * S ((cf_new_a_ec_advanced_trace) + (((cf_new_b_ec_advanced_trace) + (cf_head_ec_advanced_trace)) * S ((cf_new_b_ec_advanced_trace) + (cf_head_ec_advanced_trace)) + ((cf_head_ec_advanced_trace) + (cf_head_ec_advanced_trace)))) + ((((cf_new_b_ec_advanced_trace) + (cf_head_ec_advanced_trace)) * S ((cf_new_b_ec_advanced_trace) + (cf_head_ec_advanced_trace)) + ((cf_head_ec_advanced_trace) + (cf_head_ec_advanced_trace))) + (((cf_new_b_ec_advanced_trace) + (cf_head_ec_advanced_trace)) * S ((cf_new_b_ec_advanced_trace) + (cf_head_ec_advanced_trace)) + ((cf_head_ec_advanced_trace) + (cf_head_ec_advanced_trace))))) = S ((S (S cf_index_ec_advanced_trace)) * ec_scale_advanced)) /\ exists ff_q_cf_ec_advanced_trace_following_state. ec_history_advanced = ff_q_cf_ec_advanced_trace_following_state * S ((S (S cf_index_ec_advanced_trace)) * ec_scale_advanced) + (((cf_new_a_ec_advanced_trace) + (((cf_new_b_ec_advanced_trace) + (cf_head_ec_advanced_trace)) * S ((cf_new_b_ec_advanced_trace) + (cf_head_ec_advanced_trace)) + ((cf_head_ec_advanced_trace) + (cf_head_ec_advanced_trace)))) * S ((cf_new_a_ec_advanced_trace) + (((cf_new_b_ec_advanced_trace) + (cf_head_ec_advanced_trace)) * S ((cf_new_b_ec_advanced_trace) + (cf_head_ec_advanced_trace)) + ((cf_head_ec_advanced_trace) + (cf_head_ec_advanced_trace)))) + ((((cf_new_b_ec_advanced_trace) + (cf_head_ec_advanced_trace)) * S ((cf_new_b_ec_advanced_trace) + (cf_head_ec_advanced_trace)) + ((cf_head_ec_advanced_trace) + (cf_head_ec_advanced_trace))) + (((cf_new_b_ec_advanced_trace) + (cf_head_ec_advanced_trace)) * S ((cf_new_b_ec_advanced_trace) + (cf_head_ec_advanced_trace)) + ((cf_head_ec_advanced_trace) + (cf_head_ec_advanced_trace))))))) /\ (cf_new_b_ec_advanced_trace = cf_old_a_ec_advanced_trace /\ (cf_new_a_ec_advanced_trace = cf_new_b_ec_advanced_trace * cf_quotient_ec_advanced_trace + cf_old_b_ec_advanced_trace /\ ((exists ff_lt_cf_ec_advanced_trace_remainder. ff_lt_cf_ec_advanced_trace_remainder + S cf_old_b_ec_advanced_trace = cf_new_b_ec_advanced_trace) /\ (cf_head_ec_advanced_trace = S ((cf_quotient_ec_advanced_trace + cf_tail_ec_advanced_trace) * S (cf_quotient_ec_advanced_trace + cf_tail_ec_advanced_trace) + (cf_tail_ec_advanced_trace + cf_tail_ec_advanced_trace))))))))))) /\ ((((exists ec_gcd_left_advanced_result. a = g * ec_gcd_left_advanced_result) /\ (exists ec_gcd_right_advanced_result. b = g * ec_gcd_right_advanced_result)) /\ forall ec_gcd_common_advanced_result. (exists ec_gcd_common_left_advanced_result. a = ec_gcd_common_advanced_result * ec_gcd_common_left_advanced_result) -> (exists ec_gcd_common_right_advanced_result. b = ec_gcd_common_advanced_result * ec_gcd_common_right_advanced_result) -> exists ec_gcd_greatest_advanced_result. g = ec_gcd_common_advanced_result * ec_gcd_greatest_advanced_result)))The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.
Direct definition dependencies
Definitions depending on this notation
none