euclidean_anchored_execution_state_correct

Exact expanded first-order arithmetic statement

forall a b g l. (exists egt_list_egt_anchored egt_history_egt_anchored egt_scale_egt_anchored. ((exists cf_gcd_egt_egt_anchored_trace. ((((exists ff_h_cf_egt_egt_anchored_trace_initial_state. ff_h_cf_egt_egt_anchored_trace_initial_state + S (((cf_gcd_egt_egt_anchored_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_egt_egt_anchored_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_egt_anchored)) /\ exists ff_q_cf_egt_egt_anchored_trace_initial_state. egt_history_egt_anchored = ff_q_cf_egt_egt_anchored_trace_initial_state * S ((S (0)) * egt_scale_egt_anchored) + (((cf_gcd_egt_egt_anchored_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_egt_egt_anchored_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_egt_anchored_trace_terminal_state. ff_h_cf_egt_egt_anchored_trace_terminal_state + S (((a) + (((b) + (egt_list_egt_anchored)) * S ((b) + (egt_list_egt_anchored)) + ((egt_list_egt_anchored) + (egt_list_egt_anchored)))) * S ((a) + (((b) + (egt_list_egt_anchored)) * S ((b) + (egt_list_egt_anchored)) + ((egt_list_egt_anchored) + (egt_list_egt_anchored)))) + ((((b) + (egt_list_egt_anchored)) * S ((b) + (egt_list_egt_anchored)) + ((egt_list_egt_anchored) + (egt_list_egt_anchored))) + (((b) + (egt_list_egt_anchored)) * S ((b) + (egt_list_egt_anchored)) + ((egt_list_egt_anchored) + (egt_list_egt_anchored))))) = S ((S (l)) * egt_scale_egt_anchored)) /\ exists ff_q_cf_egt_egt_anchored_trace_terminal_state. egt_history_egt_anchored = ff_q_cf_egt_egt_anchored_trace_terminal_state * S ((S (l)) * egt_scale_egt_anchored) + (((a) + (((b) + (egt_list_egt_anchored)) * S ((b) + (egt_list_egt_anchored)) + ((egt_list_egt_anchored) + (egt_list_egt_anchored)))) * S ((a) + (((b) + (egt_list_egt_anchored)) * S ((b) + (egt_list_egt_anchored)) + ((egt_list_egt_anchored) + (egt_list_egt_anchored)))) + ((((b) + (egt_list_egt_anchored)) * S ((b) + (egt_list_egt_anchored)) + ((egt_list_egt_anchored) + (egt_list_egt_anchored))) + (((b) + (egt_list_egt_anchored)) * S ((b) + (egt_list_egt_anchored)) + ((egt_list_egt_anchored) + (egt_list_egt_anchored))))))) /\ forall cf_index_egt_egt_anchored_trace. (exists ff_lt_cf_egt_egt_anchored_trace_index. ff_lt_cf_egt_egt_anchored_trace_index + S cf_index_egt_egt_anchored_trace = l) -> exists cf_old_a_egt_egt_anchored_trace cf_old_b_egt_egt_anchored_trace cf_tail_egt_egt_anchored_trace cf_new_a_egt_egt_anchored_trace cf_new_b_egt_egt_anchored_trace cf_head_egt_egt_anchored_trace cf_quotient_egt_egt_anchored_trace. ((((exists ff_h_cf_egt_egt_anchored_trace_previous_state. ff_h_cf_egt_egt_anchored_trace_previous_state + S (((cf_old_a_egt_egt_anchored_trace) + (((cf_old_b_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)) * S ((cf_old_b_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)) + ((cf_tail_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)))) * S ((cf_old_a_egt_egt_anchored_trace) + (((cf_old_b_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)) * S ((cf_old_b_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)) + ((cf_tail_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)))) + ((((cf_old_b_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)) * S ((cf_old_b_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)) + ((cf_tail_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace))) + (((cf_old_b_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)) * S ((cf_old_b_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)) + ((cf_tail_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace))))) = S ((S (cf_index_egt_egt_anchored_trace)) * egt_scale_egt_anchored)) /\ exists ff_q_cf_egt_egt_anchored_trace_previous_state. egt_history_egt_anchored = ff_q_cf_egt_egt_anchored_trace_previous_state * S ((S (cf_index_egt_egt_anchored_trace)) * egt_scale_egt_anchored) + (((cf_old_a_egt_egt_anchored_trace) + (((cf_old_b_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)) * S ((cf_old_b_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)) + ((cf_tail_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)))) * S ((cf_old_a_egt_egt_anchored_trace) + (((cf_old_b_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)) * S ((cf_old_b_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)) + ((cf_tail_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)))) + ((((cf_old_b_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)) * S ((cf_old_b_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)) + ((cf_tail_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace))) + (((cf_old_b_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)) * S ((cf_old_b_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace)) + ((cf_tail_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace))))))) /\ ((((exists ff_h_cf_egt_egt_anchored_trace_following_state. ff_h_cf_egt_egt_anchored_trace_following_state + S (((cf_new_a_egt_egt_anchored_trace) + (((cf_new_b_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)) * S ((cf_new_b_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)) + ((cf_head_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)))) * S ((cf_new_a_egt_egt_anchored_trace) + (((cf_new_b_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)) * S ((cf_new_b_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)) + ((cf_head_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)))) + ((((cf_new_b_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)) * S ((cf_new_b_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)) + ((cf_head_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace))) + (((cf_new_b_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)) * S ((cf_new_b_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)) + ((cf_head_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace))))) = S ((S (S cf_index_egt_egt_anchored_trace)) * egt_scale_egt_anchored)) /\ exists ff_q_cf_egt_egt_anchored_trace_following_state. egt_history_egt_anchored = ff_q_cf_egt_egt_anchored_trace_following_state * S ((S (S cf_index_egt_egt_anchored_trace)) * egt_scale_egt_anchored) + (((cf_new_a_egt_egt_anchored_trace) + (((cf_new_b_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)) * S ((cf_new_b_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)) + ((cf_head_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)))) * S ((cf_new_a_egt_egt_anchored_trace) + (((cf_new_b_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)) * S ((cf_new_b_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)) + ((cf_head_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)))) + ((((cf_new_b_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)) * S ((cf_new_b_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)) + ((cf_head_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace))) + (((cf_new_b_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)) * S ((cf_new_b_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace)) + ((cf_head_egt_egt_anchored_trace) + (cf_head_egt_egt_anchored_trace))))))) /\ (cf_new_b_egt_egt_anchored_trace = cf_old_a_egt_egt_anchored_trace /\ (cf_new_a_egt_egt_anchored_trace = cf_new_b_egt_egt_anchored_trace * cf_quotient_egt_egt_anchored_trace + cf_old_b_egt_egt_anchored_trace /\ ((exists ff_lt_cf_egt_egt_anchored_trace_remainder. ff_lt_cf_egt_egt_anchored_trace_remainder + S cf_old_b_egt_egt_anchored_trace = cf_new_b_egt_egt_anchored_trace) /\ (cf_head_egt_egt_anchored_trace = S ((cf_quotient_egt_egt_anchored_trace + cf_tail_egt_egt_anchored_trace) * S (cf_quotient_egt_egt_anchored_trace + cf_tail_egt_egt_anchored_trace) + (cf_tail_egt_egt_anchored_trace + cf_tail_egt_egt_anchored_trace))))))))))) /\ ((((exists ff_h_cf_egt_egt_anchored_initial_state. ff_h_cf_egt_egt_anchored_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_egt_anchored)) /\ exists ff_q_cf_egt_egt_anchored_initial_state. egt_history_egt_anchored = ff_q_cf_egt_egt_anchored_initial_state * S ((S (0)) * egt_scale_egt_anchored) + (((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_egt_anchored_result. a = g * ec_gcd_left_egt_egt_anchored_result) /\ (exists ec_gcd_right_egt_egt_anchored_result. b = g * ec_gcd_right_egt_egt_anchored_result)) /\ forall ec_gcd_common_egt_egt_anchored_result. (exists ec_gcd_common_left_egt_egt_anchored_result. a = ec_gcd_common_egt_egt_anchored_result * ec_gcd_common_left_egt_egt_anchored_result) -> (exists ec_gcd_common_right_egt_egt_anchored_result. b = ec_gcd_common_egt_egt_anchored_result * ec_gcd_common_right_egt_egt_anchored_result) -> exists ec_gcd_greatest_egt_egt_anchored_result. g = ec_gcd_common_egt_egt_anchored_result * ec_gcd_greatest_egt_egt_anchored_result))))) -> exists s h e. ((exists cf_gcd_egt_projected. ((((exists ff_h_cf_egt_projected_initial_state. ff_h_cf_egt_projected_initial_state + S (((cf_gcd_egt_projected) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_egt_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_egt_projected_initial_state. h = ff_q_cf_egt_projected_initial_state * S ((S (0)) * e) + (((cf_gcd_egt_projected) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_egt_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_egt_projected_terminal_state. ff_h_cf_egt_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_egt_projected_terminal_state. h = ff_q_cf_egt_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_egt_projected. (exists ff_lt_cf_egt_projected_index. ff_lt_cf_egt_projected_index + S cf_index_egt_projected = l) -> exists cf_old_a_egt_projected cf_old_b_egt_projected cf_tail_egt_projected cf_new_a_egt_projected cf_new_b_egt_projected cf_head_egt_projected cf_quotient_egt_projected. ((((exists ff_h_cf_egt_projected_previous_state. ff_h_cf_egt_projected_previous_state + S (((cf_old_a_egt_projected) + (((cf_old_b_egt_projected) + (cf_tail_egt_projected)) * S ((cf_old_b_egt_projected) + (cf_tail_egt_projected)) + ((cf_tail_egt_projected) + (cf_tail_egt_projected)))) * S ((cf_old_a_egt_projected) + (((cf_old_b_egt_projected) + (cf_tail_egt_projected)) * S ((cf_old_b_egt_projected) + (cf_tail_egt_projected)) + ((cf_tail_egt_projected) + (cf_tail_egt_projected)))) + ((((cf_old_b_egt_projected) + (cf_tail_egt_projected)) * S ((cf_old_b_egt_projected) + (cf_tail_egt_projected)) + ((cf_tail_egt_projected) + (cf_tail_egt_projected))) + (((cf_old_b_egt_projected) + (cf_tail_egt_projected)) * S ((cf_old_b_egt_projected) + (cf_tail_egt_projected)) + ((cf_tail_egt_projected) + (cf_tail_egt_projected))))) = S ((S (cf_index_egt_projected)) * e)) /\ exists ff_q_cf_egt_projected_previous_state. h = ff_q_cf_egt_projected_previous_state * S ((S (cf_index_egt_projected)) * e) + (((cf_old_a_egt_projected) + (((cf_old_b_egt_projected) + (cf_tail_egt_projected)) * S ((cf_old_b_egt_projected) + (cf_tail_egt_projected)) + ((cf_tail_egt_projected) + (cf_tail_egt_projected)))) * S ((cf_old_a_egt_projected) + (((cf_old_b_egt_projected) + (cf_tail_egt_projected)) * S ((cf_old_b_egt_projected) + (cf_tail_egt_projected)) + ((cf_tail_egt_projected) + (cf_tail_egt_projected)))) + ((((cf_old_b_egt_projected) + (cf_tail_egt_projected)) * S ((cf_old_b_egt_projected) + (cf_tail_egt_projected)) + ((cf_tail_egt_projected) + (cf_tail_egt_projected))) + (((cf_old_b_egt_projected) + (cf_tail_egt_projected)) * S ((cf_old_b_egt_projected) + (cf_tail_egt_projected)) + ((cf_tail_egt_projected) + (cf_tail_egt_projected))))))) /\ ((((exists ff_h_cf_egt_projected_following_state. ff_h_cf_egt_projected_following_state + S (((cf_new_a_egt_projected) + (((cf_new_b_egt_projected) + (cf_head_egt_projected)) * S ((cf_new_b_egt_projected) + (cf_head_egt_projected)) + ((cf_head_egt_projected) + (cf_head_egt_projected)))) * S ((cf_new_a_egt_projected) + (((cf_new_b_egt_projected) + (cf_head_egt_projected)) * S ((cf_new_b_egt_projected) + (cf_head_egt_projected)) + ((cf_head_egt_projected) + (cf_head_egt_projected)))) + ((((cf_new_b_egt_projected) + (cf_head_egt_projected)) * S ((cf_new_b_egt_projected) + (cf_head_egt_projected)) + ((cf_head_egt_projected) + (cf_head_egt_projected))) + (((cf_new_b_egt_projected) + (cf_head_egt_projected)) * S ((cf_new_b_egt_projected) + (cf_head_egt_projected)) + ((cf_head_egt_projected) + (cf_head_egt_projected))))) = S ((S (S cf_index_egt_projected)) * e)) /\ exists ff_q_cf_egt_projected_following_state. h = ff_q_cf_egt_projected_following_state * S ((S (S cf_index_egt_projected)) * e) + (((cf_new_a_egt_projected) + (((cf_new_b_egt_projected) + (cf_head_egt_projected)) * S ((cf_new_b_egt_projected) + (cf_head_egt_projected)) + ((cf_head_egt_projected) + (cf_head_egt_projected)))) * S ((cf_new_a_egt_projected) + (((cf_new_b_egt_projected) + (cf_head_egt_projected)) * S ((cf_new_b_egt_projected) + (cf_head_egt_projected)) + ((cf_head_egt_projected) + (cf_head_egt_projected)))) + ((((cf_new_b_egt_projected) + (cf_head_egt_projected)) * S ((cf_new_b_egt_projected) + (cf_head_egt_projected)) + ((cf_head_egt_projected) + (cf_head_egt_projected))) + (((cf_new_b_egt_projected) + (cf_head_egt_projected)) * S ((cf_new_b_egt_projected) + (cf_head_egt_projected)) + ((cf_head_egt_projected) + (cf_head_egt_projected))))))) /\ (cf_new_b_egt_projected = cf_old_a_egt_projected /\ (cf_new_a_egt_projected = cf_new_b_egt_projected * cf_quotient_egt_projected + cf_old_b_egt_projected /\ ((exists ff_lt_cf_egt_projected_remainder. ff_lt_cf_egt_projected_remainder + S cf_old_b_egt_projected = cf_new_b_egt_projected) /\ (cf_head_egt_projected = S ((cf_quotient_egt_projected + cf_tail_egt_projected) * S (cf_quotient_egt_projected + cf_tail_egt_projected) + (cf_tail_egt_projected + cf_tail_egt_projected))))))))))) /\ (((exists ff_h_cf_egt_projected_state. ff_h_cf_egt_projected_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)) * e)) /\ exists ff_q_cf_egt_projected_state. h = ff_q_cf_egt_projected_state * S ((S (0)) * e) + (((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))))))))