EuclideanAnchoredExecution(a,b,g,k)

Hygienic expanded first-order definition

exists egt_list_transport egt_history_transport egt_scale_transport. ((exists cf_gcd_egt_transport_trace. ((((exists ff_h_cf_egt_transport_trace_initial_state. ff_h_cf_egt_transport_trace_initial_state + S (((cf_gcd_egt_transport_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_egt_transport_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_transport)) /\ exists ff_q_cf_egt_transport_trace_initial_state. egt_history_transport = ff_q_cf_egt_transport_trace_initial_state * S ((S (0)) * egt_scale_transport) + (((cf_gcd_egt_transport_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_egt_transport_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_transport_trace_terminal_state. ff_h_cf_egt_transport_trace_terminal_state + S (((a) + (((b) + (egt_list_transport)) * S ((b) + (egt_list_transport)) + ((egt_list_transport) + (egt_list_transport)))) * S ((a) + (((b) + (egt_list_transport)) * S ((b) + (egt_list_transport)) + ((egt_list_transport) + (egt_list_transport)))) + ((((b) + (egt_list_transport)) * S ((b) + (egt_list_transport)) + ((egt_list_transport) + (egt_list_transport))) + (((b) + (egt_list_transport)) * S ((b) + (egt_list_transport)) + ((egt_list_transport) + (egt_list_transport))))) = S ((S (k)) * egt_scale_transport)) /\ exists ff_q_cf_egt_transport_trace_terminal_state. egt_history_transport = ff_q_cf_egt_transport_trace_terminal_state * S ((S (k)) * egt_scale_transport) + (((a) + (((b) + (egt_list_transport)) * S ((b) + (egt_list_transport)) + ((egt_list_transport) + (egt_list_transport)))) * S ((a) + (((b) + (egt_list_transport)) * S ((b) + (egt_list_transport)) + ((egt_list_transport) + (egt_list_transport)))) + ((((b) + (egt_list_transport)) * S ((b) + (egt_list_transport)) + ((egt_list_transport) + (egt_list_transport))) + (((b) + (egt_list_transport)) * S ((b) + (egt_list_transport)) + ((egt_list_transport) + (egt_list_transport))))))) /\ forall cf_index_egt_transport_trace. (exists ff_lt_cf_egt_transport_trace_index. ff_lt_cf_egt_transport_trace_index + S cf_index_egt_transport_trace = k) -> exists cf_old_a_egt_transport_trace cf_old_b_egt_transport_trace cf_tail_egt_transport_trace cf_new_a_egt_transport_trace cf_new_b_egt_transport_trace cf_head_egt_transport_trace cf_quotient_egt_transport_trace. ((((exists ff_h_cf_egt_transport_trace_previous_state. ff_h_cf_egt_transport_trace_previous_state + S (((cf_old_a_egt_transport_trace) + (((cf_old_b_egt_transport_trace) + (cf_tail_egt_transport_trace)) * S ((cf_old_b_egt_transport_trace) + (cf_tail_egt_transport_trace)) + ((cf_tail_egt_transport_trace) + (cf_tail_egt_transport_trace)))) * S ((cf_old_a_egt_transport_trace) + (((cf_old_b_egt_transport_trace) + (cf_tail_egt_transport_trace)) * S ((cf_old_b_egt_transport_trace) + (cf_tail_egt_transport_trace)) + ((cf_tail_egt_transport_trace) + (cf_tail_egt_transport_trace)))) + ((((cf_old_b_egt_transport_trace) + (cf_tail_egt_transport_trace)) * S ((cf_old_b_egt_transport_trace) + (cf_tail_egt_transport_trace)) + ((cf_tail_egt_transport_trace) + (cf_tail_egt_transport_trace))) + (((cf_old_b_egt_transport_trace) + (cf_tail_egt_transport_trace)) * S ((cf_old_b_egt_transport_trace) + (cf_tail_egt_transport_trace)) + ((cf_tail_egt_transport_trace) + (cf_tail_egt_transport_trace))))) = S ((S (cf_index_egt_transport_trace)) * egt_scale_transport)) /\ exists ff_q_cf_egt_transport_trace_previous_state. egt_history_transport = ff_q_cf_egt_transport_trace_previous_state * S ((S (cf_index_egt_transport_trace)) * egt_scale_transport) + (((cf_old_a_egt_transport_trace) + (((cf_old_b_egt_transport_trace) + (cf_tail_egt_transport_trace)) * S ((cf_old_b_egt_transport_trace) + (cf_tail_egt_transport_trace)) + ((cf_tail_egt_transport_trace) + (cf_tail_egt_transport_trace)))) * S ((cf_old_a_egt_transport_trace) + (((cf_old_b_egt_transport_trace) + (cf_tail_egt_transport_trace)) * S ((cf_old_b_egt_transport_trace) + (cf_tail_egt_transport_trace)) + ((cf_tail_egt_transport_trace) + (cf_tail_egt_transport_trace)))) + ((((cf_old_b_egt_transport_trace) + (cf_tail_egt_transport_trace)) * S ((cf_old_b_egt_transport_trace) + (cf_tail_egt_transport_trace)) + ((cf_tail_egt_transport_trace) + (cf_tail_egt_transport_trace))) + (((cf_old_b_egt_transport_trace) + (cf_tail_egt_transport_trace)) * S ((cf_old_b_egt_transport_trace) + (cf_tail_egt_transport_trace)) + ((cf_tail_egt_transport_trace) + (cf_tail_egt_transport_trace))))))) /\ ((((exists ff_h_cf_egt_transport_trace_following_state. ff_h_cf_egt_transport_trace_following_state + S (((cf_new_a_egt_transport_trace) + (((cf_new_b_egt_transport_trace) + (cf_head_egt_transport_trace)) * S ((cf_new_b_egt_transport_trace) + (cf_head_egt_transport_trace)) + ((cf_head_egt_transport_trace) + (cf_head_egt_transport_trace)))) * S ((cf_new_a_egt_transport_trace) + (((cf_new_b_egt_transport_trace) + (cf_head_egt_transport_trace)) * S ((cf_new_b_egt_transport_trace) + (cf_head_egt_transport_trace)) + ((cf_head_egt_transport_trace) + (cf_head_egt_transport_trace)))) + ((((cf_new_b_egt_transport_trace) + (cf_head_egt_transport_trace)) * S ((cf_new_b_egt_transport_trace) + (cf_head_egt_transport_trace)) + ((cf_head_egt_transport_trace) + (cf_head_egt_transport_trace))) + (((cf_new_b_egt_transport_trace) + (cf_head_egt_transport_trace)) * S ((cf_new_b_egt_transport_trace) + (cf_head_egt_transport_trace)) + ((cf_head_egt_transport_trace) + (cf_head_egt_transport_trace))))) = S ((S (S cf_index_egt_transport_trace)) * egt_scale_transport)) /\ exists ff_q_cf_egt_transport_trace_following_state. egt_history_transport = ff_q_cf_egt_transport_trace_following_state * S ((S (S cf_index_egt_transport_trace)) * egt_scale_transport) + (((cf_new_a_egt_transport_trace) + (((cf_new_b_egt_transport_trace) + (cf_head_egt_transport_trace)) * S ((cf_new_b_egt_transport_trace) + (cf_head_egt_transport_trace)) + ((cf_head_egt_transport_trace) + (cf_head_egt_transport_trace)))) * S ((cf_new_a_egt_transport_trace) + (((cf_new_b_egt_transport_trace) + (cf_head_egt_transport_trace)) * S ((cf_new_b_egt_transport_trace) + (cf_head_egt_transport_trace)) + ((cf_head_egt_transport_trace) + (cf_head_egt_transport_trace)))) + ((((cf_new_b_egt_transport_trace) + (cf_head_egt_transport_trace)) * S ((cf_new_b_egt_transport_trace) + (cf_head_egt_transport_trace)) + ((cf_head_egt_transport_trace) + (cf_head_egt_transport_trace))) + (((cf_new_b_egt_transport_trace) + (cf_head_egt_transport_trace)) * S ((cf_new_b_egt_transport_trace) + (cf_head_egt_transport_trace)) + ((cf_head_egt_transport_trace) + (cf_head_egt_transport_trace))))))) /\ (cf_new_b_egt_transport_trace = cf_old_a_egt_transport_trace /\ (cf_new_a_egt_transport_trace = cf_new_b_egt_transport_trace * cf_quotient_egt_transport_trace + cf_old_b_egt_transport_trace /\ ((exists ff_lt_cf_egt_transport_trace_remainder. ff_lt_cf_egt_transport_trace_remainder + S cf_old_b_egt_transport_trace = cf_new_b_egt_transport_trace) /\ (cf_head_egt_transport_trace = S ((cf_quotient_egt_transport_trace + cf_tail_egt_transport_trace) * S (cf_quotient_egt_transport_trace + cf_tail_egt_transport_trace) + (cf_tail_egt_transport_trace + cf_tail_egt_transport_trace))))))))))) /\ ((((exists ff_h_cf_egt_transport_initial_state. ff_h_cf_egt_transport_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_transport)) /\ exists ff_q_cf_egt_transport_initial_state. egt_history_transport = ff_q_cf_egt_transport_initial_state * S ((S (0)) * egt_scale_transport) + (((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_transport_result. a = g * ec_gcd_left_egt_transport_result) /\ (exists ec_gcd_right_egt_transport_result. b = g * ec_gcd_right_egt_transport_result)) /\ forall ec_gcd_common_egt_transport_result. (exists ec_gcd_common_left_egt_transport_result. a = ec_gcd_common_egt_transport_result * ec_gcd_common_left_egt_transport_result) -> (exists ec_gcd_common_right_egt_transport_result. b = ec_gcd_common_egt_transport_result * ec_gcd_common_right_egt_transport_result) -> exists ec_gcd_greatest_egt_transport_result. g = ec_gcd_common_egt_transport_result * ec_gcd_greatest_egt_transport_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

Checked theorems using this definition