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.
Hygienic expanded first-order definition
((e = h + h) /\ ((exists ff_b_binary_advanced_half ff_c_binary_advanced_half. ((forall ff_i_binary_advanced_half_repeat. (exists ff_lt_binary_advanced_half_repeat_bound. ff_lt_binary_advanced_half_repeat_bound + S ff_i_binary_advanced_half_repeat = h) -> (((exists ff_h_binary_advanced_half_repeat_decoded. ff_h_binary_advanced_half_repeat_decoded + S (a) = S ((S (ff_i_binary_advanced_half_repeat)) * ff_c_binary_advanced_half)) /\ exists ff_q_binary_advanced_half_repeat_decoded. ff_b_binary_advanced_half = ff_q_binary_advanced_half_repeat_decoded * S ((S (ff_i_binary_advanced_half_repeat)) * ff_c_binary_advanced_half) + (a)))) /\ (exists ff_u_binary_advanced_half_product ff_v_binary_advanced_half_product. ((((exists ff_h_binary_advanced_half_product_start. ff_h_binary_advanced_half_product_start + S (1) = S ((S (0)) * ff_v_binary_advanced_half_product)) /\ exists ff_q_binary_advanced_half_product_start. ff_u_binary_advanced_half_product = ff_q_binary_advanced_half_product_start * S ((S (0)) * ff_v_binary_advanced_half_product) + (1))) /\ ((((exists ff_h_binary_advanced_half_product_terminal. ff_h_binary_advanced_half_product_terminal + S (u) = S ((S (h)) * ff_v_binary_advanced_half_product)) /\ exists ff_q_binary_advanced_half_product_terminal. ff_u_binary_advanced_half_product = ff_q_binary_advanced_half_product_terminal * S ((S (h)) * ff_v_binary_advanced_half_product) + (u))) /\ forall ff_i_binary_advanced_half_product. (exists ff_lt_binary_advanced_half_product_bound. ff_lt_binary_advanced_half_product_bound + S ff_i_binary_advanced_half_product = h) -> exists ff_p_binary_advanced_half_product ff_r_binary_advanced_half_product ff_s_binary_advanced_half_product. ((((exists ff_h_binary_advanced_half_product_factor. ff_h_binary_advanced_half_product_factor + S (ff_p_binary_advanced_half_product) = S ((S (ff_i_binary_advanced_half_product)) * ff_c_binary_advanced_half)) /\ exists ff_q_binary_advanced_half_product_factor. ff_b_binary_advanced_half = ff_q_binary_advanced_half_product_factor * S ((S (ff_i_binary_advanced_half_product)) * ff_c_binary_advanced_half) + (ff_p_binary_advanced_half_product))) /\ ((((exists ff_h_binary_advanced_half_product_partial. ff_h_binary_advanced_half_product_partial + S (ff_r_binary_advanced_half_product) = S ((S (ff_i_binary_advanced_half_product)) * ff_v_binary_advanced_half_product)) /\ exists ff_q_binary_advanced_half_product_partial. ff_u_binary_advanced_half_product = ff_q_binary_advanced_half_product_partial * S ((S (ff_i_binary_advanced_half_product)) * ff_v_binary_advanced_half_product) + (ff_r_binary_advanced_half_product))) /\ ((((exists ff_h_binary_advanced_half_product_successor. ff_h_binary_advanced_half_product_successor + S (ff_s_binary_advanced_half_product) = S ((S (S ff_i_binary_advanced_half_product)) * ff_v_binary_advanced_half_product)) /\ exists ff_q_binary_advanced_half_product_successor. ff_u_binary_advanced_half_product = ff_q_binary_advanced_half_product_successor * S ((S (S ff_i_binary_advanced_half_product)) * ff_v_binary_advanced_half_product) + (ff_s_binary_advanced_half_product))) /\ ff_s_binary_advanced_half_product = ff_r_binary_advanced_half_product * ff_p_binary_advanced_half_product)))))))) /\ (exists ff_b_binary_advanced_full ff_c_binary_advanced_full. ((forall ff_i_binary_advanced_full_repeat. (exists ff_lt_binary_advanced_full_repeat_bound. ff_lt_binary_advanced_full_repeat_bound + S ff_i_binary_advanced_full_repeat = e) -> (((exists ff_h_binary_advanced_full_repeat_decoded. ff_h_binary_advanced_full_repeat_decoded + S (a) = S ((S (ff_i_binary_advanced_full_repeat)) * ff_c_binary_advanced_full)) /\ exists ff_q_binary_advanced_full_repeat_decoded. ff_b_binary_advanced_full = ff_q_binary_advanced_full_repeat_decoded * S ((S (ff_i_binary_advanced_full_repeat)) * ff_c_binary_advanced_full) + (a)))) /\ (exists ff_u_binary_advanced_full_product ff_v_binary_advanced_full_product. ((((exists ff_h_binary_advanced_full_product_start. ff_h_binary_advanced_full_product_start + S (1) = S ((S (0)) * ff_v_binary_advanced_full_product)) /\ exists ff_q_binary_advanced_full_product_start. ff_u_binary_advanced_full_product = ff_q_binary_advanced_full_product_start * S ((S (0)) * ff_v_binary_advanced_full_product) + (1))) /\ ((((exists ff_h_binary_advanced_full_product_terminal. ff_h_binary_advanced_full_product_terminal + S (z) = S ((S (e)) * ff_v_binary_advanced_full_product)) /\ exists ff_q_binary_advanced_full_product_terminal. ff_u_binary_advanced_full_product = ff_q_binary_advanced_full_product_terminal * S ((S (e)) * ff_v_binary_advanced_full_product) + (z))) /\ forall ff_i_binary_advanced_full_product. (exists ff_lt_binary_advanced_full_product_bound. ff_lt_binary_advanced_full_product_bound + S ff_i_binary_advanced_full_product = e) -> exists ff_p_binary_advanced_full_product ff_r_binary_advanced_full_product ff_s_binary_advanced_full_product. ((((exists ff_h_binary_advanced_full_product_factor. ff_h_binary_advanced_full_product_factor + S (ff_p_binary_advanced_full_product) = S ((S (ff_i_binary_advanced_full_product)) * ff_c_binary_advanced_full)) /\ exists ff_q_binary_advanced_full_product_factor. ff_b_binary_advanced_full = ff_q_binary_advanced_full_product_factor * S ((S (ff_i_binary_advanced_full_product)) * ff_c_binary_advanced_full) + (ff_p_binary_advanced_full_product))) /\ ((((exists ff_h_binary_advanced_full_product_partial. ff_h_binary_advanced_full_product_partial + S (ff_r_binary_advanced_full_product) = S ((S (ff_i_binary_advanced_full_product)) * ff_v_binary_advanced_full_product)) /\ exists ff_q_binary_advanced_full_product_partial. ff_u_binary_advanced_full_product = ff_q_binary_advanced_full_product_partial * S ((S (ff_i_binary_advanced_full_product)) * ff_v_binary_advanced_full_product) + (ff_r_binary_advanced_full_product))) /\ ((((exists ff_h_binary_advanced_full_product_successor. ff_h_binary_advanced_full_product_successor + S (ff_s_binary_advanced_full_product) = S ((S (S ff_i_binary_advanced_full_product)) * ff_v_binary_advanced_full_product)) /\ exists ff_q_binary_advanced_full_product_successor. ff_u_binary_advanced_full_product = ff_q_binary_advanced_full_product_successor * S ((S (S ff_i_binary_advanced_full_product)) * ff_v_binary_advanced_full_product) + (ff_s_binary_advanced_full_product))) /\ ff_s_binary_advanced_full_product = ff_r_binary_advanced_full_product * ff_p_binary_advanced_full_product))))))))))The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.
Direct definition dependencies
Definitions depending on this notation
none