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 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 hag_left_factor_egt_terminal. a = g * hag_left_factor_egt_terminal) /\ (exists hag_right_factor_egt_terminal. b = g * hag_right_factor_egt_terminal)) /\ forall hag_divisor_egt_terminal. (exists hag_common_left_egt_terminal. a = hag_divisor_egt_terminal * hag_common_left_egt_terminal) -> (exists hag_common_right_egt_terminal. b = hag_divisor_egt_terminal * hag_common_right_egt_terminal) -> exists hag_greatest_factor_egt_terminal. g = hag_divisor_egt_terminal * hag_greatest_factor_egt_terminal))Constructive proof overview
Generated structural guide
Every anchored Euclidean execution independently satisfies the full original relational greatest-common-divisor specification.
The unchanged tactic script uses 0 declared prerequisites and contains 11 exact native proof lines.
Alpha v34 checked-use · first admitted v22 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
none
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
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.
01Fix variables and assumptionsL1–5
02Separate the logical casesL6–10
03Use earlier factsL11–11
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L11
exact hanchored_witness_witness_witness_right_right