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
exists ff_u_defined_product ff_v_defined_product. ((((exists ff_h_defined_product_start. ff_h_defined_product_start + S (1) = S ((S (0)) * ff_v_defined_product)) /\ exists ff_q_defined_product_start. ff_u_defined_product = ff_q_defined_product_start * S ((S (0)) * ff_v_defined_product) + (1))) /\ ((((exists ff_h_defined_product_terminal. ff_h_defined_product_terminal + S (z) = S ((S (l)) * ff_v_defined_product)) /\ exists ff_q_defined_product_terminal. ff_u_defined_product = ff_q_defined_product_terminal * S ((S (l)) * ff_v_defined_product) + (z))) /\ forall ff_i_defined_product. (exists ff_lt_defined_product_bound. ff_lt_defined_product_bound + S ff_i_defined_product = l) -> exists ff_p_defined_product ff_r_defined_product ff_s_defined_product. ((((exists ff_h_defined_product_factor. ff_h_defined_product_factor + S (ff_p_defined_product) = S ((S (ff_i_defined_product)) * c)) /\ exists ff_q_defined_product_factor. b = ff_q_defined_product_factor * S ((S (ff_i_defined_product)) * c) + (ff_p_defined_product))) /\ ((((exists ff_h_defined_product_partial. ff_h_defined_product_partial + S (ff_r_defined_product) = S ((S (ff_i_defined_product)) * ff_v_defined_product)) /\ exists ff_q_defined_product_partial. ff_u_defined_product = ff_q_defined_product_partial * S ((S (ff_i_defined_product)) * ff_v_defined_product) + (ff_r_defined_product))) /\ ((((exists ff_h_defined_product_successor. ff_h_defined_product_successor + S (ff_s_defined_product) = S ((S (S ff_i_defined_product)) * ff_v_defined_product)) /\ exists ff_q_defined_product_successor. ff_u_defined_product = ff_q_defined_product_successor * S ((S (S ff_i_defined_product)) * ff_v_defined_product) + (ff_s_defined_product))) /\ ff_s_defined_product = ff_r_defined_product * ff_p_defined_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
Checked theorems using this definition
none directly; see definition consumers