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 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 ec_gcd_left_execution_correct. a = g * ec_gcd_left_execution_correct) /\ (exists ec_gcd_right_execution_correct. b = g * ec_gcd_right_execution_correct)) /\ forall ec_gcd_common_execution_correct. (exists ec_gcd_common_left_execution_correct. a = ec_gcd_common_execution_correct * ec_gcd_common_left_execution_correct) -> (exists ec_gcd_common_right_execution_correct. b = ec_gcd_common_execution_correct * ec_gcd_common_right_execution_correct) -> exists ec_gcd_greatest_execution_correct. g = ec_gcd_common_execution_correct * ec_gcd_greatest_execution_correct))Constructive proof overview
Generated structural guide
Every expanded Euclidean execution independently certifies its output against the original relational gcd specification.
The unchanged tactic script uses 0 declared prerequisites and contains 10 exact native proof lines.
Alpha v34 checked-use · first admitted v21 · 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 casesL6–9
03Use earlier factsL10–10
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L10
exact hexecution_witness_witness_witness_right