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. (exists pa_b_olte_two pa_c_olte_two. ((forall pa_i_olte_two_repeat. (exists pa_lt_olte_two_repeat_bound. pa_lt_olte_two_repeat_bound + S pa_i_olte_two_repeat = 2) -> (((exists pa_h_olte_two_repeat_decoded. pa_h_olte_two_repeat_decoded + S (a) = S ((S (pa_i_olte_two_repeat)) * pa_c_olte_two)) /\ exists pa_q_olte_two_repeat_decoded. pa_b_olte_two = pa_q_olte_two_repeat_decoded * S ((S (pa_i_olte_two_repeat)) * pa_c_olte_two) + (a)))) /\ (exists pa_u_olte_two_product pa_v_olte_two_product. ((((exists pa_h_olte_two_product_start. pa_h_olte_two_product_start + S (1) = S ((S (0)) * pa_v_olte_two_product)) /\ exists pa_q_olte_two_product_start. pa_u_olte_two_product = pa_q_olte_two_product_start * S ((S (0)) * pa_v_olte_two_product) + (1))) /\ ((((exists pa_h_olte_two_product_terminal. pa_h_olte_two_product_terminal + S (a * a) = S ((S (2)) * pa_v_olte_two_product)) /\ exists pa_q_olte_two_product_terminal. pa_u_olte_two_product = pa_q_olte_two_product_terminal * S ((S (2)) * pa_v_olte_two_product) + (a * a))) /\ forall pa_i_olte_two_product. (exists pa_lt_olte_two_product_bound. pa_lt_olte_two_product_bound + S pa_i_olte_two_product = 2) -> exists pa_p_olte_two_product pa_r_olte_two_product pa_s_olte_two_product. ((((exists pa_h_olte_two_product_factor. pa_h_olte_two_product_factor + S (pa_p_olte_two_product) = S ((S (pa_i_olte_two_product)) * pa_c_olte_two)) /\ exists pa_q_olte_two_product_factor. pa_b_olte_two = pa_q_olte_two_product_factor * S ((S (pa_i_olte_two_product)) * pa_c_olte_two) + (pa_p_olte_two_product))) /\ ((((exists pa_h_olte_two_product_partial. pa_h_olte_two_product_partial + S (pa_r_olte_two_product) = S ((S (pa_i_olte_two_product)) * pa_v_olte_two_product)) /\ exists pa_q_olte_two_product_partial. pa_u_olte_two_product = pa_q_olte_two_product_partial * S ((S (pa_i_olte_two_product)) * pa_v_olte_two_product) + (pa_r_olte_two_product))) /\ ((((exists pa_h_olte_two_product_successor. pa_h_olte_two_product_successor + S (pa_s_olte_two_product) = S ((S (S pa_i_olte_two_product)) * pa_v_olte_two_product)) /\ exists pa_q_olte_two_product_successor. pa_u_olte_two_product = pa_q_olte_two_product_successor * S ((S (S pa_i_olte_two_product)) * pa_v_olte_two_product) + (pa_s_olte_two_product))) /\ pa_s_olte_two_product = pa_r_olte_two_product * pa_p_olte_two_product))))))))Constructive proof overview
Generated structural guide
The relational second power is the actual square, including a zero base.
The unchanged tactic script uses 2 declared prerequisites and contains 9 exact native proof lines.
Alpha v34 checked-use · first admitted v29 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
pow_successor_compose Alpha theorem; checked-use authorized EL0008 lte_power_one_exactDirect dependents
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.
Named ingredients (1)
01Fix variables and assumptionsL1–1
Work with arbitrary variables or the premises of the current implication.
- L1
intro a
02Use earlier factsL2–8
Instantiate or apply named facts and discharge the corresponding proof obligations.
03Calculate and transport equalitiesL9–9
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L9
refl