euclidean_execution_trace_correct

forall a b g l. (exists ec_list_execution ec_history_execution ec_scale_execution. ((exists cf_gcd_ec_execution_trace. ((((exists ff_h_cf_ec_execution_trace_initial_state. ff_h_cf_ec_execution_trace_initial_state + S (((cf_gcd_ec_execution_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_ec_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)) * ec_scale_execution)) /\ exists ff_q_cf_ec_execution_trace_initial_state. ec_history_execution = ff_q_cf_ec_execution_trace_initial_state * S ((S (0)) * ec_scale_execution) + (((cf_gcd_ec_execution_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_ec_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_ec_execution_trace_terminal_state. ff_h_cf_ec_execution_trace_terminal_state + S (((a) + (((b) + (ec_list_execution)) * S ((b) + (ec_list_execution)) + ((ec_list_execution) + (ec_list_execution)))) * S ((a) + (((b) + (ec_list_execution)) * S ((b) + (ec_list_execution)) + ((ec_list_execution) + (ec_list_execution)))) + ((((b) + (ec_list_execution)) * S ((b) + (ec_list_execution)) + ((ec_list_execution) + (ec_list_execution))) + (((b) + (ec_list_execution)) * S ((b) + (ec_list_execution)) + ((ec_list_execution) + (ec_list_execution))))) = S ((S (l)) * ec_scale_execution)) /\ exists ff_q_cf_ec_execution_trace_terminal_state. ec_history_execution = ff_q_cf_ec_execution_trace_terminal_state * S ((S (l)) * ec_scale_execution) + (((a) + (((b) + (ec_list_execution)) * S ((b) + (ec_list_execution)) + ((ec_list_execution) + (ec_list_execution)))) * S ((a) + (((b) + (ec_list_execution)) * S ((b) + (ec_list_execution)) + ((ec_list_execution) + (ec_list_execution)))) + ((((b) + (ec_list_execution)) * S ((b) + (ec_list_execution)) + ((ec_list_execution) + (ec_list_execution))) + (((b) + (ec_list_execution)) * S ((b) + (ec_list_execution)) + ((ec_list_execution) + (ec_list_execution))))))) /\ forall cf_index_ec_execution_trace. (exists ff_lt_cf_ec_execution_trace_index. ff_lt_cf_ec_execution_trace_index + S cf_index_ec_execution_trace = l) -> exists cf_old_a_ec_execution_trace cf_old_b_ec_execution_trace cf_tail_ec_execution_trace cf_new_a_ec_execution_trace cf_new_b_ec_execution_trace cf_head_ec_execution_trace cf_quotient_ec_execution_trace. ((((exists ff_h_cf_ec_execution_trace_previous_state. ff_h_cf_ec_execution_trace_previous_state + S (((cf_old_a_ec_execution_trace) + (((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) * S ((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) + ((cf_tail_ec_execution_trace) + (cf_tail_ec_execution_trace)))) * S ((cf_old_a_ec_execution_trace) + (((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) * S ((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) + ((cf_tail_ec_execution_trace) + (cf_tail_ec_execution_trace)))) + ((((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) * S ((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) + ((cf_tail_ec_execution_trace) + (cf_tail_ec_execution_trace))) + (((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) * S ((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) + ((cf_tail_ec_execution_trace) + (cf_tail_ec_execution_trace))))) = S ((S (cf_index_ec_execution_trace)) * ec_scale_execution)) /\ exists ff_q_cf_ec_execution_trace_previous_state. ec_history_execution = ff_q_cf_ec_execution_trace_previous_state * S ((S (cf_index_ec_execution_trace)) * ec_scale_execution) + (((cf_old_a_ec_execution_trace) + (((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) * S ((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) + ((cf_tail_ec_execution_trace) + (cf_tail_ec_execution_trace)))) * S ((cf_old_a_ec_execution_trace) + (((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) * S ((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) + ((cf_tail_ec_execution_trace) + (cf_tail_ec_execution_trace)))) + ((((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) * S ((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) + ((cf_tail_ec_execution_trace) + (cf_tail_ec_execution_trace))) + (((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) * S ((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) + ((cf_tail_ec_execution_trace) + (cf_tail_ec_execution_trace))))))) /\ ((((exists ff_h_cf_ec_execution_trace_following_state. ff_h_cf_ec_execution_trace_following_state + S (((cf_new_a_ec_execution_trace) + (((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) * S ((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) + ((cf_head_ec_execution_trace) + (cf_head_ec_execution_trace)))) * S ((cf_new_a_ec_execution_trace) + (((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) * S ((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) + ((cf_head_ec_execution_trace) + (cf_head_ec_execution_trace)))) + ((((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) * S ((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) + ((cf_head_ec_execution_trace) + (cf_head_ec_execution_trace))) + (((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) * S ((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) + ((cf_head_ec_execution_trace) + (cf_head_ec_execution_trace))))) = S ((S (S cf_index_ec_execution_trace)) * ec_scale_execution)) /\ exists ff_q_cf_ec_execution_trace_following_state. ec_history_execution = ff_q_cf_ec_execution_trace_following_state * S ((S (S cf_index_ec_execution_trace)) * ec_scale_execution) + (((cf_new_a_ec_execution_trace) + (((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) * S ((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) + ((cf_head_ec_execution_trace) + (cf_head_ec_execution_trace)))) * S ((cf_new_a_ec_execution_trace) + (((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) * S ((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) + ((cf_head_ec_execution_trace) + (cf_head_ec_execution_trace)))) + ((((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) * S ((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) + ((cf_head_ec_execution_trace) + (cf_head_ec_execution_trace))) + (((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) * S ((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) + ((cf_head_ec_execution_trace) + (cf_head_ec_execution_trace))))))) /\ (cf_new_b_ec_execution_trace = cf_old_a_ec_execution_trace /\ (cf_new_a_ec_execution_trace = cf_new_b_ec_execution_trace * cf_quotient_ec_execution_trace + cf_old_b_ec_execution_trace /\ ((exists ff_lt_cf_ec_execution_trace_remainder. ff_lt_cf_ec_execution_trace_remainder + S cf_old_b_ec_execution_trace = cf_new_b_ec_execution_trace) /\ (cf_head_ec_execution_trace = S ((cf_quotient_ec_execution_trace + cf_tail_ec_execution_trace) * S (cf_quotient_ec_execution_trace + cf_tail_ec_execution_trace) + (cf_tail_ec_execution_trace + cf_tail_ec_execution_trace))))))))))) /\ ((((exists ec_gcd_left_execution_result. a = g * ec_gcd_left_execution_result) /\ (exists ec_gcd_right_execution_result. b = g * ec_gcd_right_execution_result)) /\ forall ec_gcd_common_execution_result. (exists ec_gcd_common_left_execution_result. a = ec_gcd_common_execution_result * ec_gcd_common_left_execution_result) -> (exists ec_gcd_common_right_execution_result. b = ec_gcd_common_execution_result * ec_gcd_common_right_execution_result) -> exists ec_gcd_greatest_execution_result. g = ec_gcd_common_execution_result * ec_gcd_greatest_execution_result)))) -> exists s h e. (exists cf_gcd_execution_projected. ((((exists ff_h_cf_execution_projected_initial_state. ff_h_cf_execution_projected_initial_state + S (((cf_gcd_execution_projected) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_execution_projected) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) + ((((0) + (0)) * S ((0) + (0)) + ((0) + (0))) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0))))) = S ((S (0)) * e)) /\ exists ff_q_cf_execution_projected_initial_state. h = ff_q_cf_execution_projected_initial_state * S ((S (0)) * e) + (((cf_gcd_execution_projected) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_execution_projected) + (((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_execution_projected_terminal_state. ff_h_cf_execution_projected_terminal_state + S (((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) * S ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) + ((((b) + (s)) * S ((b) + (s)) + ((s) + (s))) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s))))) = S ((S (l)) * e)) /\ exists ff_q_cf_execution_projected_terminal_state. h = ff_q_cf_execution_projected_terminal_state * S ((S (l)) * e) + (((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) * S ((a) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s)))) + ((((b) + (s)) * S ((b) + (s)) + ((s) + (s))) + (((b) + (s)) * S ((b) + (s)) + ((s) + (s))))))) /\ forall cf_index_execution_projected. (exists ff_lt_cf_execution_projected_index. ff_lt_cf_execution_projected_index + S cf_index_execution_projected = l) -> exists cf_old_a_execution_projected cf_old_b_execution_projected cf_tail_execution_projected cf_new_a_execution_projected cf_new_b_execution_projected cf_head_execution_projected cf_quotient_execution_projected. ((((exists ff_h_cf_execution_projected_previous_state. ff_h_cf_execution_projected_previous_state + S (((cf_old_a_execution_projected) + (((cf_old_b_execution_projected) + (cf_tail_execution_projected)) * S ((cf_old_b_execution_projected) + (cf_tail_execution_projected)) + ((cf_tail_execution_projected) + (cf_tail_execution_projected)))) * S ((cf_old_a_execution_projected) + (((cf_old_b_execution_projected) + (cf_tail_execution_projected)) * S ((cf_old_b_execution_projected) + (cf_tail_execution_projected)) + ((cf_tail_execution_projected) + (cf_tail_execution_projected)))) + ((((cf_old_b_execution_projected) + (cf_tail_execution_projected)) * S ((cf_old_b_execution_projected) + (cf_tail_execution_projected)) + ((cf_tail_execution_projected) + (cf_tail_execution_projected))) + (((cf_old_b_execution_projected) + (cf_tail_execution_projected)) * S ((cf_old_b_execution_projected) + (cf_tail_execution_projected)) + ((cf_tail_execution_projected) + (cf_tail_execution_projected))))) = S ((S (cf_index_execution_projected)) * e)) /\ exists ff_q_cf_execution_projected_previous_state. h = ff_q_cf_execution_projected_previous_state * S ((S (cf_index_execution_projected)) * e) + (((cf_old_a_execution_projected) + (((cf_old_b_execution_projected) + (cf_tail_execution_projected)) * S ((cf_old_b_execution_projected) + (cf_tail_execution_projected)) + ((cf_tail_execution_projected) + (cf_tail_execution_projected)))) * S ((cf_old_a_execution_projected) + (((cf_old_b_execution_projected) + (cf_tail_execution_projected)) * S ((cf_old_b_execution_projected) + (cf_tail_execution_projected)) + ((cf_tail_execution_projected) + (cf_tail_execution_projected)))) + ((((cf_old_b_execution_projected) + (cf_tail_execution_projected)) * S ((cf_old_b_execution_projected) + (cf_tail_execution_projected)) + ((cf_tail_execution_projected) + (cf_tail_execution_projected))) + (((cf_old_b_execution_projected) + (cf_tail_execution_projected)) * S ((cf_old_b_execution_projected) + (cf_tail_execution_projected)) + ((cf_tail_execution_projected) + (cf_tail_execution_projected))))))) /\ ((((exists ff_h_cf_execution_projected_following_state. ff_h_cf_execution_projected_following_state + S (((cf_new_a_execution_projected) + (((cf_new_b_execution_projected) + (cf_head_execution_projected)) * S ((cf_new_b_execution_projected) + (cf_head_execution_projected)) + ((cf_head_execution_projected) + (cf_head_execution_projected)))) * S ((cf_new_a_execution_projected) + (((cf_new_b_execution_projected) + (cf_head_execution_projected)) * S ((cf_new_b_execution_projected) + (cf_head_execution_projected)) + ((cf_head_execution_projected) + (cf_head_execution_projected)))) + ((((cf_new_b_execution_projected) + (cf_head_execution_projected)) * S ((cf_new_b_execution_projected) + (cf_head_execution_projected)) + ((cf_head_execution_projected) + (cf_head_execution_projected))) + (((cf_new_b_execution_projected) + (cf_head_execution_projected)) * S ((cf_new_b_execution_projected) + (cf_head_execution_projected)) + ((cf_head_execution_projected) + (cf_head_execution_projected))))) = S ((S (S cf_index_execution_projected)) * e)) /\ exists ff_q_cf_execution_projected_following_state. h = ff_q_cf_execution_projected_following_state * S ((S (S cf_index_execution_projected)) * e) + (((cf_new_a_execution_projected) + (((cf_new_b_execution_projected) + (cf_head_execution_projected)) * S ((cf_new_b_execution_projected) + (cf_head_execution_projected)) + ((cf_head_execution_projected) + (cf_head_execution_projected)))) * S ((cf_new_a_execution_projected) + (((cf_new_b_execution_projected) + (cf_head_execution_projected)) * S ((cf_new_b_execution_projected) + (cf_head_execution_projected)) + ((cf_head_execution_projected) + (cf_head_execution_projected)))) + ((((cf_new_b_execution_projected) + (cf_head_execution_projected)) * S ((cf_new_b_execution_projected) + (cf_head_execution_projected)) + ((cf_head_execution_projected) + (cf_head_execution_projected))) + (((cf_new_b_execution_projected) + (cf_head_execution_projected)) * S ((cf_new_b_execution_projected) + (cf_head_execution_projected)) + ((cf_head_execution_projected) + (cf_head_execution_projected))))))) /\ (cf_new_b_execution_projected = cf_old_a_execution_projected /\ (cf_new_a_execution_projected = cf_new_b_execution_projected * cf_quotient_execution_projected + cf_old_b_execution_projected /\ ((exists ff_lt_cf_execution_projected_remainder. ff_lt_cf_execution_projected_remainder + S cf_old_b_execution_projected = cf_new_b_execution_projected) /\ (cf_head_execution_projected = S ((cf_quotient_execution_projected + cf_tail_execution_projected) * S (cf_quotient_execution_projected + cf_tail_execution_projected) + (cf_tail_execution_projected + cf_tail_execution_projected)))))))))))