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.
Statement with defined notation
∀ x. Pow(2,2,x) → x = 4Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.
Definitions used by this theorem
In the theorem statement
1 occurrences
In local proof propositions
0 occurrences
Exact expanded native-PA statement
forall x. (exists pa_b_bie_two_two pa_c_bie_two_two. ((forall pa_i_bie_two_two_repeat. (exists pa_lt_bie_two_two_repeat_bound. pa_lt_bie_two_two_repeat_bound + S pa_i_bie_two_two_repeat = 2) -> (((exists pa_h_bie_two_two_repeat_decoded. pa_h_bie_two_two_repeat_decoded + S (2) = S ((S (pa_i_bie_two_two_repeat)) * pa_c_bie_two_two)) /\ exists pa_q_bie_two_two_repeat_decoded. pa_b_bie_two_two = pa_q_bie_two_two_repeat_decoded * S ((S (pa_i_bie_two_two_repeat)) * pa_c_bie_two_two) + (2)))) /\ (exists pa_u_bie_two_two_product pa_v_bie_two_two_product. ((((exists pa_h_bie_two_two_product_start. pa_h_bie_two_two_product_start + S (1) = S ((S (0)) * pa_v_bie_two_two_product)) /\ exists pa_q_bie_two_two_product_start. pa_u_bie_two_two_product = pa_q_bie_two_two_product_start * S ((S (0)) * pa_v_bie_two_two_product) + (1))) /\ ((((exists pa_h_bie_two_two_product_terminal. pa_h_bie_two_two_product_terminal + S (x) = S ((S (2)) * pa_v_bie_two_two_product)) /\ exists pa_q_bie_two_two_product_terminal. pa_u_bie_two_two_product = pa_q_bie_two_two_product_terminal * S ((S (2)) * pa_v_bie_two_two_product) + (x))) /\ forall pa_i_bie_two_two_product. (exists pa_lt_bie_two_two_product_bound. pa_lt_bie_two_two_product_bound + S pa_i_bie_two_two_product = 2) -> exists pa_p_bie_two_two_product pa_r_bie_two_two_product pa_s_bie_two_two_product. ((((exists pa_h_bie_two_two_product_factor. pa_h_bie_two_two_product_factor + S (pa_p_bie_two_two_product) = S ((S (pa_i_bie_two_two_product)) * pa_c_bie_two_two)) /\ exists pa_q_bie_two_two_product_factor. pa_b_bie_two_two = pa_q_bie_two_two_product_factor * S ((S (pa_i_bie_two_two_product)) * pa_c_bie_two_two) + (pa_p_bie_two_two_product))) /\ ((((exists pa_h_bie_two_two_product_partial. pa_h_bie_two_two_product_partial + S (pa_r_bie_two_two_product) = S ((S (pa_i_bie_two_two_product)) * pa_v_bie_two_two_product)) /\ exists pa_q_bie_two_two_product_partial. pa_u_bie_two_two_product = pa_q_bie_two_two_product_partial * S ((S (pa_i_bie_two_two_product)) * pa_v_bie_two_two_product) + (pa_r_bie_two_two_product))) /\ ((((exists pa_h_bie_two_two_product_successor. pa_h_bie_two_two_product_successor + S (pa_s_bie_two_two_product) = S ((S (S pa_i_bie_two_two_product)) * pa_v_bie_two_two_product)) /\ exists pa_q_bie_two_two_product_successor. pa_u_bie_two_two_product = pa_q_bie_two_two_product_successor * S ((S (S pa_i_bie_two_two_product)) * pa_v_bie_two_two_product) + (pa_s_bie_two_two_product))) /\ pa_s_bie_two_two_product = pa_r_bie_two_two_product * pa_p_bie_two_two_product)))))))) -> x = 4Proof neighborhood
Direct theorem prerequisites
Direct theorem dependents
Definition-aware tactic body
Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.
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.